For example, $P_1^1 = sin\theta$ and $P_2^2 = 3 sin^2\theta$ seem to be not orthogonal to each other because the integral $$\int_0^\pi (sin\theta)(3 sin^2\theta) sin\theta d\theta$$ is not zero. Am I missing something? Thank you for your help!
Because the orthogonality is supplied by the azimuthal direction when the two $z$-components of the angular momentum differ. This is why we usually bundle them together into the spherical harmonics instead of handling them separately.